Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
58 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Add and Subtract Polynomials2.1Goal p Add and subtract polynomials.Georgia
PerformanceStandard(s)
MM1A2c
Your Notes
VOCABULARY
Monomial
Degree of a monomial
Polynomial
Degree of a polynomial
Leading coefficient
Binomial
Trinomial
Write 7 1 2x4 2 4x so that the exponents decrease from left to right. Identify the degree and the leading coefficient of the polynomial.
SolutionConsider the degree of each of the polynomial's terms.
Degree is . Degree is . Degree is .
7 1 2x4 2 4x
The polynomial can be written as . The greatest degree is , so the degree of the polynomial is , and the leading coefficient is .
Example 1 Rewrite a polynomial
ga1nt_02.indd 31 4/16/07 9:24:28 AM
ga1_ntg_02.indd 58ga1_ntg_02.indd 58 4/20/07 11:45:04 AM4/20/07 11:45:04 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 59
Your Notes Checkpoint Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient of the polynomial.
1. 5x 1 13 1 8x3
2. 4y4 2 7y5 1 2y
3. 25m2 1 1 2 9m
4. 2r3 1 4r4 1 r 2 5r2
Find the sum (a) (4x3 1 x2 2 5) 1 (7x 1 x3 2 3x2)and (b) (x2 1 x 1 8) 1 (x2 2 x 2 1).
Solutiona. Vertical format: Align like 4x3 1 x2 2 5 terms in vertical columns. 1 x3 2 3x2 1 7x
b. Horizontal format: Group like terms and simplify.
(x2 1 x 1 8) 1 (x2 2 x 2 1) 5 ( ) 1 ( ) 1 ( )
5
Example 2 Add polynomials
If a particular power of the variable appears in one polynomial but not the other, leave a space in that column, or write the term with a coefficient of 0.
ga1nt_02.indd 32 4/16/07 9:24:30 AM
ga1_ntg_02.indd 59ga1_ntg_02.indd 59 4/20/07 11:45:06 AM4/20/07 11:45:06 AM
60 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Homework
5. (3x4 2 2x2 2 1) 1 (5x3 2 x2 1 9x4)
6. (4x2 2 15 1 6x3) 1 (8x 1 24 1 x2)
Checkpoint Find the sum.
Find the difference (a) (4z2 2 3) 2 (22z2 1 5z 2 1)and (b) (3x2 1 6x 2 4) 2 (x2 2 x 2 7).
Solutiona. ( 4z2 2 3) 4z2 2 3
2 (22z2 1 5z 2 1) 2z2 5z 1
b. (3x2 1 6x 2 4) 2 (x2 2 x 2 7)
5 3x2 1 6x 2 4
5
5
Example 3 Subtract polynomials
Remember to multiply each term in the polynomial by 21 when you write the subtraction as addition.
7. (3t2 2 5t 1 t4) 2 (11t4 2 3t2)
8. (4p3 1 3p 2 6) 2 (22p3 2 p2 1 3)
Checkpoint Find the difference.
ga1nt_02.indd 33 4/16/07 9:24:30 AM
ga1_ntg_02.indd 60ga1_ntg_02.indd 60 4/20/07 11:45:08 AM4/20/07 11:45:08 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 61
Name ——————————————————————— Date ————————————
LESSON
2.1 PracticeWrite the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coeffi cient of the polynomial.
1. 8n6 2. 29z 1 1 3. 4 1 2x5
4. 18x 2 x2 1 2 5. 3y3 1 4y2 1 8 6. m 2 20m3 1 5
7. 28 1 10a4 2 3a7 8. 4z 1 z3 2 5z2 1 6z4 9. 8h3 2 6h4 1 h7
Tell whether the expression is a polynomial. If it is a polynomial, fi nd its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial.
10. 6m2 11. 3x 12. y22 1 4
13. 3b2 2 2 14. 1 } 2 x2 2 2x 1 1 15. 6x3 2 1.4x
Find the sum or difference.
16. (6x 1 4) 1 (x 1 5) 17. (4m2 2 5) 1 (3m2 2 2)
18. (2y2 1 y 2 1) 1 (7y2 1 4y 2 3) 19. (3x2 1 5) 2 (x2 1 2)
ga1_ntg_02.indd 61ga1_ntg_02.indd 61 4/20/07 11:45:09 AM4/20/07 11:45:09 AM
62 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
20. (10a2 1 4a 2 5) 2 (3a2 1 2a 1 1) 21. (m2 2 3m 1 4) 2 (2m2 1 5m 1 1)
Write a polynomial that represents the perimeter of the fi gure.
22.
x 1 2 x 1 1
2x 1 1
23.
x 1 1
x 1 4
x 1 5
x 2 1
24. Library Books For 1997 through 2007, the number F of fi ction books (in ten thousands) and the number N of nonfi ction books (in ten thousands) borrowed from a library can be modeled by
F 5 0.01t2 1 0.08t 1 7 and N 5 0.004t2 1 0.05t 1 5
where t is the number of years since 1997. Write an equation for the total number B of fi ction and nonfi ction books borrowed from the library in a year from 1997 to 2007.
25. Photograph Mat A mat in a frame has an opening
a � 3b � 4x � 2
xNot drawn to scale
4x
for a photograph as shown in the fi gure. Find the area of the mat if the area of the opening is given by A 5 πab. Leave your answer in terms of π.
LESSON
2.1 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 62ga1_ntg_02.indd 62 4/20/07 11:45:10 AM4/20/07 11:45:10 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 63
2.2 Multiply PolynomialsGeorgiaPerformanceStandard(s)
MM1A2c, MM1A2g
Your Notes
VOCABULARY
Area model for polynomial arithmetic
Volume model for polynomial arithmetic
Goal p Multiply polynomials.
Find the product 3x3(2x3 2 x2 2 7x 2 3).
Solution
3x3(2x3 2 x2 2 7x 2 3)5 3x3( ) 2 3x3( ) 2 3x3( ) 2 3x3( )
5 2 2 2
Example 1 Multiply a monomial and a polynomial
Write a polynomial for the
2x � 5
x � 1x
1
1
x
x2
x
xx
x2 x
1
1
x
1
1
x
1
1
x
1
1
area of the model shown.
SolutionYou know that the area of a rectangle is the product of its length and width. In the model, let represent the length and let represent the width. To find the total area of the model, the areas of each rectangular part.
A 5 l 3 w 5 1 1 1 1 1
1 1 1 1 1
1 1 1
5 1 1
Example 2 Multiply polynomials using an area model
ga1nt_02.indd 34 4/16/07 9:24:32 AM
ga1_ntg_02.indd 63ga1_ntg_02.indd 63 4/20/07 11:45:11 AM4/20/07 11:45:11 AM
64 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
1. Find the product (2x2)(x3 2 5x2 1 3x 2 7).
2. The dimensions of a rectangle are 4x 1 1 and x 1 2. Draw an area model. Then write an expression for the area of the rectangle.
Checkpoint Complete the following exercises.
Find the product (3b2 2 2b 1 5)(5b 2 6).
Solution(3b2 2 2b 1 5)(5b 2 6)
5 (5b 2 6) 2 (5b 2 6)
1 (5b 2 6)
5
5
Example 3 Multiply polynomials horizontally
ga1nt_02.indd 35 4/16/07 9:24:33 AM
ga1_ntg_02.indd 64ga1_ntg_02.indd 64 4/20/07 11:45:12 AM4/20/07 11:45:12 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 65
Your Notes
3. (a2 1 5a 2 4)(2a 1 3)
4. (m 1 3)(5m 2 4)
5. (2k 2 3)(7k 2 8)
Checkpoint Find the product.
Find the product (2c 1 7)(c 2 9).
Solution
(2c 1 7)(c 2 9)
5 2c( ) 1 2c( ) 1 7( ) 1 7( )
5
5
Example 4 Multiply binomials
Remember that the terms of (c 2 9)are c and 29. They are not c and 9.
ga1nt_02.indd 36 4/16/07 9:24:34 AM
ga1_ntg_02.indd 65ga1_ntg_02.indd 65 4/20/07 11:45:14 AM4/20/07 11:45:14 AM
66 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Write a polynomial for the volume
x � 6
x � 3
x
of the rectangular prism shown.
SolutionYou know that the volume of a rectangular prism is the product of its , , and . In the figure shown, let represent the length, let represent the width, and let represent the height.
Volume 5 length p width p height
5 ( )( )
5 [ ( ) 1 ( ) 1 ( )
1 ( )]
5 ( 1 1 1 )
5 ( )
5
5
Example 5 Multiply polynomials using a volume model
Homework
6. Write an expression for the x � 7
3x � 4x
volume of the prism shown.
Checkpoint Complete the following exercise.
ga1nt_02.indd 37 4/16/07 9:24:34 AM
ga1_ntg_02.indd 66ga1_ntg_02.indd 66 4/20/07 11:45:15 AM4/20/07 11:45:15 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 67
Name ——————————————————————— Date ————————————
LESSON
2.2 PracticeFind the product.
1. x(3x2 2 2x 1 1) 2. 2y(3y3 1 y2 2 4) 3. 23m(m2 1 4m 2 1)
4. d2(4d2 2 3d 1 1) 5. 2w3(w2 1 3w) 6. 2a2(a2 1 3a 2 1)
7. (4a 1 1)(2a 2 1) 8. (w 1 1)(w2 1 2w 1 1) 9. (m 2 2)(m2 2 2m 1 3)
10. (y 2 3)(8y 1 1) 11. (5b 2 1)(3b 1 2) 12. (2d 2 4)(3d 2 1)
13. (3x 1 1)(2x 1 2) 14. (6x 2 2)(x 1 4) 15. (2s 2 5)(s 1 3)
Simplify the expression.
16. p( p3 1 2p) 1 ( p 2 3)( p 1 5)
17. (x 1 3)(x 1 8) 2 x(2x 1 4)
18. (r 2 6)(r 2 2) 1 (r 1 4)(r 2 9)
ga1_ntg_02.indd 67ga1_ntg_02.indd 67 4/20/07 11:45:17 AM4/20/07 11:45:17 AM
68 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Write a polynomial for the area of the model.
19.
8x
4
x
20.
5x3
6
x
21. Volume You have come up with a plan for building
(4x 1 8) in.
(3x 1 6) in.
24 in.a wooden box to hold all of your sports equipment as shown.
a. Write a polynomial that represents the volume of the box.
b. Find the volume of the box when x 5 10.
22. National Park System During the period 1990–2002, the number A of acres (in thousands) making up the national park system in the United States and the percent P (in decimal form) of this amount that is parks can be modeled by
A 5 211t 1 76,226
and
P 5 20.0008t2 1 0.009t 1 0.6
where t is the number of years since 1990.
a. Find the values of A and P for t 5 0. What does the product A p P mean for t 5 0 in the context of this problem?
b. Write an equation that models the number of acres (in thousands) that are just parks as a function of the number of years since 1990.
LESSON
2.2 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 68ga1_ntg_02.indd 68 4/20/07 11:45:17 AM4/20/07 11:45:17 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 69
2.3 Find Special Products of Polynomials
GeorgiaPerformanceStandard(s)
MM1A2c
Your Notes
Goal p Use special product patterns to multiply binomials.
SQUARE OF A BINOMIAL PATTERN
Algebra
(a 1 b)2 5 a2 1 b2
(a 2 b)2 5 a2 1 b2
Example
(x 1 4)2 5 x2 1 16
(3x 2 2)2 5 9x2 1 4
Find the product.
Solution
a. (4x 1 3)2 5 (4x)2 1 32
5 16x2 1 9
b. (3x 2 5y)2 5 (3x)2 1 (5y)2
5 9x2 1 25y2
Example 1 Use the square of a binomial pattern
1. (x 1 9)2
2. (2x 2 7)2
3. (5r 1 s)2
Checkpoint Find the product.
When you use special product patterns, remember that a and b can be numbers, variables, or variable expressions.
ga1nt_02.indd 38 4/16/07 9:24:36 AM
ga1_ntg_02.indd 69ga1_ntg_02.indd 69 4/20/07 11:45:18 AM4/20/07 11:45:18 AM
70 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your NotesSUM AND DIFFERENCE PATTERN
Algebra(a 1 b)(a 2 b) 5 2 2 2
Example(x 1 4)(x 2 4) 5 2 2
Find the product.
Solution
a. (n 1 3)(n 2 3) 5 2 2 2 Sum and difference pattern
5 2 2 Simplify.
b. (4x 1 y)(4x 2 y) 5 2 2 2 Sum and difference pattern
5 2 2 2 Simplify.
Example 2 Use the sum and difference pattern
Use special products to find the product 17 p 23.
Solution
Notice that 17 is 3 less than while 23 is 3 more than .
17 p 23 5 ( 2 3)( 1 3) Write as product.
5 Sum and difference pattern
5 Evaluate powers.
5 Simplify.
Example 3 Use special products and mental math
ga1nt_02.indd 39 4/16/07 9:24:37 AM
ga1_ntg_02.indd 70ga1_ntg_02.indd 70 4/20/07 11:45:20 AM4/20/07 11:45:20 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 71
Your Notes
4. Find the product (z 1 6)(z 2 6).
5. Find the product (4x 1 3)(4x 2 3).
6. Find the product (x 1 5y)(x 2 5y).
7. Describe how you can use special products to find 392.
Checkpoint Complete the following exercises.
Homework
ga1nt_02.indd 40 4/16/07 9:24:37 AM
ga1_ntg_02.indd 71ga1_ntg_02.indd 71 4/20/07 11:45:21 AM4/20/07 11:45:21 AM
72 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Find the missing term.
1. (a 2 b)2 5 a2 2 ? 1 b2 2. (m 1 n)2 5 m2 1 ? 1 n2
3. (x 2 1)2 5 x2 2 ? 1 1 4. (x 1 5)2 5 x2 1 ? 1 25
5. (x 2 y)(x 1 y) 5 x2 2 ? 6. (x 2 3)(x 1 3) 5 x2 2 ?
Match the product with its polynomial.
7. (2x 1 3)(2x 2 3) 8. (2x 1 3)2 9. (2x 2 3)2
A. 4x2 1 12x 1 9 B. 4x2 2 12x 1 9 C. 4x2 2 9
Find the product of the square of the binomial.
10. (x 1 4)2 11. (m 2 8)2 12. (a 1 10)2
13. (p 2 12)2 14. (2y 1 1)2 15. (3y 2 1)2
16. (10r 2 1)2 17. (4n 1 2)2 18. (3c 2 2)2
Find the product of the sum and difference.
19. (z 1 5)(z 2 5) 20. (b 2 2)(b 1 2) 21. (n 2 8)(n 1 8)
LESSON
2.3 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 72ga1_ntg_02.indd 72 4/20/07 11:45:23 AM4/20/07 11:45:23 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 73
Name ——————————————————————— Date ————————————
LESSON
2.3 Practice continued
22. (a 1 10)(a 2 10) 23. (2x 1 1)(2x 2 1) 24. (5m 2 1)(5m 1 1)
25. (4d 1 1)(4d 2 1) 26. (3p 1 2)(3p 2 2) 27. (2r 2 3)(2r 1 3)
Describe how you can use mental math to fi nd the product.
28. 13 p 7 29. 24 p 36 30. 51 p 69
31. Total Profi t For 1997 through 2007, the number N of units (in thousands) produced by a manufacturing plant can be modeled by N 5 3t 1 2 and the profi t per unit P (in dollars) can be modeled by P 5 3t 2 2 where t is the number of years since 1997. Write a polynomial that models the total profi t T (in thousands of dollars).
32. Eye Color In humans, the brown eye gene B is dominant Mother
Fath
er
B
B
b
b
BB Bb
bB bb
and the blue eye gene b is recessive. This means that humans whose eye genes are BB, Bb, or bB have brown eyes and those with bb have blue eyes. The Punnett square at the right shows the results of eye colors for children of parents who each have one B gene and one b gene.
a. Write a polynomial that models the percent of possible gene combinations of a child.
b. What percent of the possible gene combinations results in a child with blue eyes?
ga1_ntg_02.indd 73ga1_ntg_02.indd 73 4/20/07 11:45:23 AM4/20/07 11:45:23 AM
74 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.4 Use the Binomial TheoremGoal p Use the Binomial Theorem to expand binomials.Georgia
PerformanceStandard(s)
MM1A2d
Your Notes
VOCABULARY
Pascal's triangle
Pascal's Triangle
1 n 5 0 (0th row)
1 1 n 5 1 (1st row)
1 1 n 5 2 (2nd row)
1 1 n 5 3 (3rd row)
1 1 n 5 4 (4th row)
The first and last numbers in each row are .Beginning with the second row, every other number is formed by the two numbers immediately above the number.
Binomial expansion
(a 1 b)0 5 1
(a 1 b)1 5 1a 1 1b
(a 1 b)2 5 1a2 1 1 1b2
(a 1 b)3 5 1a3 1 1 1 1b3
(a 1 b)4 5 1a4 1 1 1 1 1b4
ga1nt_02.indd 41 4/18/07 1:53:37 PM
ga1_ntg_02.indd 74ga1_ntg_02.indd 74 4/20/07 11:45:24 AM4/20/07 11:45:24 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 75
Your Notes
1. Find the numbers in the eighth row of Pascal's triangle.
Checkpoint Complete the following exercise.
Use the Binomial Theorem and Pascal's triangle to write the binomial expansion of (x 1 5)4.
SolutionThe binomial coefficients from the fourth row of Pascal's triangle are , , , , and .So, the expansion is as follows.
(x 1 5)4 5 (x4) 1 (x3)(5) 1 (x2)(5)2
1 (x)(5)3 1 (5)4
5
Example 2 Expand a power of a binomial sum
Use the fourth row of Pascal's triangle to find the numbers in the fifth and sixth rows of Pascal's triangle.
SolutionWrite the fifth row of Pascal's triangle by adding numbers from the row. Write the sixth row of Pascal's triangle by adding numbers from the row.
n 5 4 (4th row) 1 4 6 4 1
n 5 5 (5th row) 1 1
n 5 6 (6th row) 1 1
Example 1 Use Pascal's triangle
ga1nt_02.indd 42 4/16/07 9:24:39 AM
ga1_ntg_02.indd 75ga1_ntg_02.indd 75 4/20/07 11:45:25 AM4/20/07 11:45:25 AM
76 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
2. (x 1 3)5
3. (4 1 y)4
Checkpoint Use the Binomial Theorem and Pascal's triangle to write the binomial expansion.
Use the Binomial Theorem and Pascal's triangle to write the binomial expansion of (x 2 6)3.
SolutionThe binomial coefficients from the third row of Pascal's triangle are . So, the expansion is as follows.
(x 2 6)3 5 [x 1 ( )]3
5 ( )3 1 ( )2( )
1 ( )( )2 1 ( )3
5
Example 3 Expand a power of a binomial difference
4. (x 2 7)4
5. (5 2 a)3
Checkpoint Use the Binomial Theorem and Pascal's triangle to write the binomial expansion.
ga1nt_02.indd 43 4/16/07 9:24:40 AM
ga1_ntg_02.indd 76ga1_ntg_02.indd 76 4/20/07 11:45:27 AM4/20/07 11:45:27 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 77
Your Notes
Use the Binomial Theorem and Pascal's triangle to write the binomial expansion of (5 1 2x)4.
SolutionThe binomial coefficients from the fourth row of Pascal's triangle are . So, the expansion is as follows.
(5 1 2x)4 5 ( )4 1 ( )3( )
1 ( )2( )2 1 ( )( )3
1 ( )4
5
Example 4 Expand a power of a binomial sum
6. (8 2 5y)4
Checkpoint Use the Binomial Theorem and Pascal's triangle to write the binomial expansion.
Find the coefficient of x3 in (4x 1 3)4.
SolutionThe binomial coefficients from the fourth row of Pascal's triangle are . So, the expansion is as follows.
(4x 1 3)4 5 ( )4 1 ( )3 ( )
1 ( )2( )2 1 ( )( )3
1 ( )4
The coefficient of the x3-term is ( )( )3( ) 5 .
Example 5 Find a coefficient in an expansion
7. Find the coefficient of x2 in the expansion of (7 2 x)5.
Checkpoint Complete the following exercise.
Homework
ga1nt_02.indd 44 4/16/07 9:24:41 AM
ga1_ntg_02.indd 77ga1_ntg_02.indd 77 4/20/07 11:45:28 AM4/20/07 11:45:28 AM
78 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
1. Find the numbers in the tenth row of Pascal’s triangle.
Use the Binomial Theorem and Pascal’s triangle to write the binomial expansion.
2. (x 1 6)3 3. (4 1 m)5
4. ( y 1 2)4 5. (k 1 1)6
6. (5 1 w)4 7. (8 1 j)3
8. (a 2 3)2 9. (6 2 r)4
10. (10 2 s)3 11. (c 2 8)4
12. (2 2 z)5 13. ( p 2 5)3
LESSON
2.4 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 78ga1_ntg_02.indd 78 4/20/07 11:45:30 AM4/20/07 11:45:30 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 79
Name ——————————————————————— Date ————————————
LESSON
2.4 Practice continued
14. (3x 2 4)3 15. (5 1 2y)4
16. (2k 1 7)3 17. (8 2 5n)5
18. (4u 1 v)3 19. (c 2 4d )4
20. Find the coeffi cient of x2 in the expansion of (x 1 3)4.
21. Find the coeffi cient of x3 in the expansion of (2x 2 9)5.
22. Find the coeffi cient of x4 in the expansion of (5x 1 4)6.
23. Error Analysis Describe and correct the error in writing the binomial expansion.
(3 1 2y)3 5 27 1 27y 1 9y2 1 y3 X
ga1_ntg_02.indd 79ga1_ntg_02.indd 79 4/20/07 11:45:30 AM4/20/07 11:45:30 AM
80 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.5 Solve Polynomial Equations in Factored FormGoal p Solve polynomial equations.Georgia
PerformanceStandard(s)
MM1A2f
Your Notes
VOCABULARY
Roots
Vertical motion model
ZERO-PRODUCT PROPERTY
Let a and b be real numbers. If ab 5 0, then 5 0 or 5 0.
Solve (x 2 5)(x 1 4) 5 0.
Solution (x 2 5)(x 1 4) 5 0 Write original
equation.
5 0 or 5 0 property
x 5 or x 5 Solve for x.
The solutions of the equation are .
CHECK Substitute each solution into the original equation to check.
( 2 5)( 1 4) 0 0 ( 2 5)( 1 4) 0 0
0 0 0 0
5 0 5 0
Example 1 Use the zero-product property
ga1nt_02.indd 45 4/16/07 9:24:42 AM
ga1_ntg_02.indd 80ga1_ntg_02.indd 80 4/20/07 11:45:31 AM4/20/07 11:45:31 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 81
Your Notes
1. (x 1 6)(x 2 3) 5 0
2. (x 2 8)(x 2 5) 5 0
Checkpoint Solve the equation.
Solve 3x2 1 15x 5 0.
Solution
3x2 1 15x 5 0 Write original equation.
5 0 Factor left side.
5 0 or 5 0 Zero-product property
x 5 or x 5 Solve for x.
The solutions of the equation are .
Example 2 Solve an equation by factoring
Solve 9b2 5 24b.
Solution
9b2 5 24b Write original equation.
5 0 Subtract from each side.
5 0 Factor left side.
5 0 or 5 0 Zero-product property
b 5 or b 5 Solve for b.
The solutions of the equation are .
Example 3 Solve an equation by factoring
To use the zero-product property, you must write the equation so that one side is 0. For this reason, must be subtracted from each side of the equation.
ga1nt_02.indd 46 4/16/07 9:24:43 AM
ga1_ntg_02.indd 81ga1_ntg_02.indd 81 4/20/07 11:45:32 AM4/20/07 11:45:32 AM
82 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Fountain A fountain sprays water from the ground into the air with an initial vertical velocity of 20 feet per second. After how many seconds does it land on the ground?
SolutionStep 1 Write a model for the water's height above ground.
h 5 216t2 1 vt 1 s Vertical motion model
h 5 216t2 1 t 1 v 5 and s 5
h 5 216t2 1 Simplify.
Step 2 Substitute for h. When the water lands, its height above the ground is feet. Solve for t.
5 216t2 1 Substitute for h.
5 Factor right side.
or Zero-product property
or Solve for t.
The water lands on the ground seconds after it is sprayed.
Example 4 Solve a multi-step problem
3. Solve d2 2 7d 5 0. 4. Solve 8b2 5 2b.
5. In Example 4, suppose the initial vertical velocity is 18 feet per second. After how many seconds does the water land on the ground?
Checkpoint Complete the following exercises.
Homework
The solution t 5 0 means that before the water is sprayed, its height above the ground is 0 feet.
ga1nt_02.indd 47 4/16/07 9:24:44 AM
ga1_ntg_02.indd 82ga1_ntg_02.indd 82 4/20/07 11:45:34 AM4/20/07 11:45:34 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 83
Name ——————————————————————— Date ————————————
LESSON
2.5 PracticeMatch the equation with its solutions.
1. (x 1 4)(x 1 5) 5 0 2. (x 2 4)(x 1 5) 5 0 3. (x 2 5)(x 2 4) 5 0
A. 25 and 4 B. 25 and 24 C. 4 and 5
Solve the equation.
4. (x 1 6)(x 1 2) 5 0 5. (p 2 5)(p 1 3) 5 0 6. (b 2 7)(b 2 10) 5 0
7. (m 2 8)(m 1 1) 5 0 8. (a 2 9)(a 1 9) 5 0 9. (y 1 15)(y 1 12) 5 0
10. (c 2 25)(c 1 50) 5 0 11. (2z 2 2)(z 1 3) 5 0 12. (2n 2 6)(n 2 2) 5 0
Match the equation with its solutions.
13. 4a2 1 a 5 0 14. a2 1 4a 5 0 15. a2 2 4a 5 0
A. 0 and 4 B. 0 and 24 C. 0 and 2 1 } 4
ga1_ntg_02.indd 83ga1_ntg_02.indd 83 4/20/07 11:45:35 AM4/20/07 11:45:35 AM
84 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Solve the equation.
16. a2 1 8a 5 0 17. n2 2 7n 5 0 18. 2w2 1 2w 5 0
19. 3p2 2 3p 5 0 20. 4c2 2 8c 5 0 21. 5x2 1 10x 5 0
22. 15m2 5 23m 23. 24r2 5 42r 24. 28k2 5 32k
25. Hot Air Balloon An object is dropped from a hot-air balloon 1296 feet above the ground. The height of the object is given by
h 5 216(t 2 9)(t 1 9)
where the height h is measured in feet, and the time t is measured in seconds. After how many seconds will the object hit the ground?
26. Kickball A kickball is kicked upward with an initial vertical velocity of 3.2 meters per second. The height of the ball is given by
h 5 29.8t2 1 3.2t
where the height h is measured in feet, and the time t is measured in seconds. After how many seconds does the ball land?
LESSON
2.5 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 84ga1_ntg_02.indd 84 4/20/07 11:45:36 AM4/20/07 11:45:36 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 85
2.6 Factor x2 1 bx 1 cGoal p Factor trinomials of the form x2 1 bx 1 c.Georgia
PerformanceStandard(s)
MM1A2f, MM1A3a, MM1A3c
Your Notes
FACTORING x2 1 bx 1 c
Algebra
x2 1 bx 1 c 5 (x 1 p)(x 1 q) provided 5 band 5 c.
Example
x2 1 6x 1 5 5 ( )( ) because 5 6and 5 5.
Factor x2 1 10x 1 16.
Solution
Find two factors of whose sum is .Make an organized list.
Factors of Sum of factors
16, 16 1 5
8, 8 1 5
4, 4 1 5
The factors 8 and have a sum of , so they are the correct values of p and q.
x2 1 10x 1 16 5 (x 1 8)( )
CHECK(x 1 8)( ) 5 Multiply.
5 Simplify.
Example 1 Factor when b and c are positive
ga1nt_02.indd 48 4/16/07 9:24:45 AM
ga1_ntg_02.indd 85ga1_ntg_02.indd 85 4/20/07 11:45:36 AM4/20/07 11:45:36 AM
86 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Factor a2 2 5a 1 6.
SolutionBecause b is negative and c is positive, p and qmust .
Factors of Sum of factors
1 ( ) 5
1 ( ) 5
a2 2 5a 1 6 5 ( )( )
Example 2 Factor when b is negative and c is positive
Factor y2 1 3y 2 10.
Solution
Because c is negative, p and q must .
Factors of Sum of factors
210, 210 1 5
10, 10 1 5
25, 25 1 5
5, 5 1 5
y2 1 3y 2 10 5 ( )( )
Example 3 Factor when b is positive and c is negative
1. x 1 7x 1 12 2. x 1 9x 1 8
Checkpoint Factor the trinomial.
ga1nt_02.indd 49 4/16/07 9:24:46 AM
ga1_ntg_02.indd 86ga1_ntg_02.indd 86 4/20/07 11:45:38 AM4/20/07 11:45:38 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 87
Your Notes
Solve the equation x2 1 7x 5 18.
x2 1 7x 5 18 Write original equation.
x2 1 7x 2 5 0 Subtract from each side.
5 0 Factor left side.
or Zero-product property
or Solve for x.
The solutions of the equation are .
Example 4 Solve a polynomial equation
3. x2 2 12x 1 27 4. x2 2 9x 1 20
5. y2 1 4y 2 21 6. z2 1 2z 2 24
Checkpoint Factor the trinomial.
7. Solve the equation s2 2 12s 5 13.
Checkpoint Complete the following exercise.Homework
ga1nt_02.indd 50 4/16/07 9:24:47 AM
ga1_ntg_02.indd 87ga1_ntg_02.indd 87 4/20/07 11:45:39 AM4/20/07 11:45:39 AM
88 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Match the trinomial with its correct factorization.
1. x2 2 4x 2 12 2. x2 2 x 2 12 3. x2 1 4x 2 12
A. (x 1 6)(x 2 2) B. (x 2 6)(x 1 2) C. (x 1 3)(x 2 4)
Factor the trinomial.
4. x2 1 6x 1 5 5. a2 1 10a 1 21 6. w2 1 8w 1 15
7. p2 2 3p 2 10 8. c2 1 10c 2 11 9. y2 1 5y 2 14
10. n2 2 4n 1 3 11. b2 2 5b 1 6 12. r2 2 12r 1 35
13. z2 1 7z 1 12 14. s2 2 3s 2 18 15. d2 2 5d 2 24
Solve the equation.
16. x2 1 5x 1 4 5 0 17. d2 1 7d 1 10 5 0 18. p2 1 9p 1 14 5 0
19. w2 2 12w 1 11 5 0 20. n2 2 n 2 6 5 0 21. a2 2 12a 1 35 5 0
LESSON
2.6 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 88ga1_ntg_02.indd 88 4/20/07 11:45:41 AM4/20/07 11:45:41 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 89
Name ——————————————————————— Date ————————————
LESSON
2.6 Practice continued
22. y2 2 4y 2 5 5 0 23. m2 1 2m 2 15 5 0 24. b2 1 6b 2 7 5 0
25. w(w 1 1) 5 12 26. x(x 2 3) 5 10 27. m(m 2 5) 5 6
28. b(b 1 4) 5 21 29. p(p 1 5) 5 36 30. r(r 2 3) 5 4
31. Boardwalk A boardwalk is being built along two sides
x ft
80 ft
120 ftx ft
of a beach area. The beach area is rectangular with a width of 80 feet and a length of 120 feet. The boardwalk will have the same width on each side of the beach area. If the combined area of the beach and the boardwalk is 16,500 square feet, then the area can be modeled by (x 1 80)(x 1 120) 5 16,500. How wide should the boardwalk be?
32. Note Board Design You are designing a note board
1.5 ft(x 1 1) ft
x ftCorkboardDry
eraseboard
that is made of corkboard and dry erase board. The area of the corkboard is 6 square feet.
a. Write an equation for the area of the corkboard.
b. Find the dimensions of the corkboard.
c. Find the area of the dry erase board.
ga1_ntg_02.indd 89ga1_ntg_02.indd 89 4/20/07 11:45:41 AM4/20/07 11:45:41 AM
90 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Factor ax2 1 bx 1 c2.7GeorgiaPerformanceStandard(s)
MM1A2f
Your Notes
Goal p Factor trinomials of the form ax2 1 bx 1 c.
Factor 2x2 2 11x 1 5.
SolutionBecause b is negative and c is positive, both factors of c must be . You must consider the of the factors of 5, because the x-terms of the possible factorizations are different.
Factors of 2
Factors of 5
Possible factorization
Middle term whenmultiplied
1, 2 21, (x 2 1)(2x ) 2 2x 5
1, 2 25, (x 2 5)(2x ) 2 10x 5
2x2 2 11x 1 5 5 (x 2 )(2x )
Example 1 Factor when b is negative and c is positive
Factor 5n2 1 2n 2 3.
SolutionBecause b is positive and c is negative, the factors of c have .
Factors of 5
Factors of 23
Possible factorization
Middle term whenmultiplied
1, 5 1, (n 1 1)(5n )
1, 5 21, (n 2 1)(5n )
1, 5 3, (n 1 3)(5n )
1, 5 23, (n 2 3)(5n )
5n2 1 2n 2 3 5 (n )(5n )
Example 2 Factor when b is positive and c is negative
ga1nt_02.indd 51 4/16/07 9:24:48 AM
ga1_ntg_02.indd 90ga1_ntg_02.indd 90 4/20/07 11:45:42 AM4/20/07 11:45:42 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 91
Your Notes
Factor 24x2 1 4x 1 3.
Solution
Step 1 Factor from each term of the trinomial.
24x2 1 4x 1 3 5 ( )
Step 2 Factor the trinomial . Because c is , the factors of c must have
.
Factors of 4
Factors of 23
Possible factorization
Middle term whenmultiplied
1, 4 1, (x 1 1)(4x )
1, 4 3, (x 1 3)(4x )
1, 4 21, (x 2 1)(4x )
1, 4 23, (x 2 3)(4x )
2, 2 1, (2x 1 1)(2x )
2, 2 21, (2x 2 1)(2x )
24x2 1 4x 1 3 5
Example 3 Factor when a is negative
1. 3x2 2 5x 1 2 2. 2m2 1 m 2 21
Checkpoint Factor the trinomial.
Remember to include the that you factored out in Step 1.
3. Factor 22y2 2 11y 2 5.
Checkpoint Complete the following exercise.Homework
ga1nt_02.indd 52 4/16/07 9:24:49 AM
ga1_ntg_02.indd 91ga1_ntg_02.indd 91 4/20/07 11:45:44 AM4/20/07 11:45:44 AM
92 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Match the trinomial with its correct factorization.
1. 4x2 2 2x 2 2 2. 4x2 2 7x 2 2 3. 4x2 1 7x 2 2
A. (4x 1 1)(x 2 2) B. (2x 1 1)(2x 2 2) C. (4x 2 1)(x 1 2)
Factor the trinomial.
4. 2x2 2 2x 1 15 5. 2m2 1 3m 2 2 6. 2p2 1 5p 1 14
7. 2w2 1 7w 1 3 8. 3y2 1 5y 1 2 9. 2b2 1 b 2 1
10. 3n2 2 3 11. 5a2 1 13a 2 6 12. 2z2 1 9z 2 5
13. 7d2 2 15d 1 2 14. 2r2 2 12r 1 10 15. 6s2 2 13s 1 2
Solve the equation.
16. 2x2 1 7x 2 15 5 0 17. 3n2 1 13n 1 4 5 0 18. 4b2 1 2b 2 2 5 0
19. 2m2 1 5m 2 3 5 0 20. 3p2 1 11p 2 4 5 0 21. 3y2 1 11y 1 10 5 0
22. 4r2 1 8r 1 3 5 0 23. 9w2 1 3w 2 2 5 0 24. 5a2 2 8a 2 4 5 0
LESSON
2.7 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 92ga1_ntg_02.indd 92 4/20/07 11:45:45 AM4/20/07 11:45:45 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 93
Name ——————————————————————— Date ————————————
LESSON
2.7 Practice continued
25. 3c2 1 19c 2 14 5 0 26. 8z2 1 6z 1 1 5 0 27. 12d2 1 14d 2 6 5 0
Find the zeros of the polynomial function.
28. f (x) 5 2x2 2 4x 1 5 29. g(x) 5 3x2 2 13x 2 10 30. h(x) 5 22x2 1 9x 1 5
31. g(x) 5 2x2 1 5x 2 6 32. f (x) 5 4x2 2 9x 1 2 33. g(x) 5 22x2 2 9x 1 18
34. h(x) 5 2x2 1 7x 2 4 35. h(x) 5 6x2 1 3x 2 9 36. f (x) 5 24x2 2 9x 2 2
37. Ball Toss A ball is tossed into the air from a height of 8 feet with an initial velocity of 8 feet per second. Find the time t (in seconds) it takes for the object to reach the ground by solving the equation 216t2 1 8t 1 8 5 0.
38. Wallpaper You trimmed a large strip of wallpaper from a scrap
(x 2 6) in.
4x in.
x in.
(4x 2 15) in.
to fi t into the corner of a wall you are wallpapering. You trimmed 15 inches from the length and 6 inches from the width. The area of the resulting strip of wallpaper is 684 square inches.
a. If the length of the original strip of wallpaper is four times the original width, write a polynomial that represents the area of the trimmed strip of wallpaper.
b. What are the dimensions of the original scrap of wallpaper?
ga1_ntg_02.indd 93ga1_ntg_02.indd 93 4/20/07 11:45:46 AM4/20/07 11:45:46 AM
94 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.8 Factor Special ProductsGoal p Factor special products.Georgia
PerformanceStandard(s)
MM1A2f
Your Notes
DIFFERENCE OF TWO SQUARES PATTERN
Algebra
a2 2 b2 5 (a 1 b)( )
Example
9x2 2 4 5 (3x)2 2 22 5 ( )( )
Factor the polynomial.
a. z2 2 81 5 z2 2 2
5 (z 1 )(z 2 )
b. 16x2 2 9 5 ( )2 2 2
5 ( 1 )( 2 )
c. a2 2 25b2 5 a2 2 ( )2
5 (a 1 )(a 2 )
d. 4 2 16n2 5 ( 2 )
5 [( )2 2 ( )2]
5 ( 1 )( 2 )
Example 1 Factor the differences of two squares
1. x2 2 100 2. 49y2 2 25
3. c2 2 9d2 4. 45 2 80m2
Checkpoint Factor the polynomial.
ga1nt_02.indd 53 4/16/07 9:24:50 AM
ga1_ntg_02.indd 94ga1_ntg_02.indd 94 4/20/07 11:45:47 AM4/20/07 11:45:47 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 95
Your NotesPERFECT SQUARE TRINOMIAL PATTERN
Algebra
a2 1 2ab 1 b2 5 ( )2
a2 2 2ab 1 b2 5 ( )2
Example
x2 1 8x 1 16 5 x2 1 2(x p 4) 1 42 5 ( )2
x2 2 6x 1 9 5 x2 2 2(x p 3) 1 32 5 ( )2
Factor the polynomial.
a. x2 2 10x 1 25 5 x2 2 2( )( ) 1 2
5 ( )2
b. y2 1 12y 1 36 5 y2 1 2( )( ) 1 2
5 ( )2
Example 2 Factor perfect square trinomials
5. x2 1 14x 1 49 6. t2 2 22t 1 121
Checkpoint Factor the polynomial.
Factor the polynomial.
a. 4y2 2 12y 1 9 5 ( )2 2 2( ) 1 2
5 ( )2
b. 23z2 1 24z 2 48 5 (z2 2 8z 1 16)
5 [z2 2 2( ) 1 2]
5 ( )2
c. 49s2 1 56st 1 16t2 5 ( )2 1 2( ) 1 2
5 ( )2
Example 3 Factor perfect square trinomials
ga1nt_02.indd 54 4/16/07 9:24:51 AM
ga1_ntg_02.indd 95ga1_ntg_02.indd 95 4/20/07 11:45:49 AM4/20/07 11:45:49 AM
96 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Solve the equation x2 1 x 11}4
5 0.
x2 1 x 11}4 5 0 Write original equation.
5 0 Multiply each side by .
5 0 Write left side as a2 1 2ab 1 b2.
5 0 Perfect square trinomial pattern
5 0 Zero-product property
x 5 Solve for x.
Example 4 Solve a polynomial equation
This equation has two identical solutions, because it has two identical factors.
7. 16x2 2 40xy 1 25y2 8. 25r2 2 20r 2 20
Checkpoint Factor the polynomial.
9. m2 2 8m 1 16 5 0
10. t2 2 121 5 0
Checkpoint Solve the equation.
Homework
ga1nt_02.indd 55 4/16/07 9:24:52 AM
ga1_ntg_02.indd 96ga1_ntg_02.indd 96 4/20/07 11:45:50 AM4/20/07 11:45:50 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 97
Name ——————————————————————— Date ————————————
LESSON
2.8 PracticeMatch the trinomial with its correct factorization.
1. x2 2 25 2. x2 1 10x 1 25 3. x2 2 10x 1 25
A. (x 1 5)2 B. (x 2 5)(x 1 5) C. (x 2 5)2
Factor the difference of two squares.
4. x2 2 1 5. b2 2 81 6. m2 2 100
7. p2 2 225 8. 4y2 2 1 9. 16n2 2 25
10. 4r2 2 121 11. 9s2 2 144 12. c2 2 625
Factor the perfect square trinomial.
13. x2 1 6x 1 9 14. b2 1 10b 1 25 15. w2 2 12w 1 36
16. m2 2 8m 1 16 17. r2 2 20r 1 100 18. z2 1 16z 1 64
19. s2 1 22s 1 121 20. x2 2 16x 1 64 21. 4c2 1 4c 1 1
22. 16d2 1 8d 1 1 23. 9y2 2 6y 1 1 24. 9p2 2 12p 1 4
25. 4m2 1 28mn 1 49n2 26. 100x2 2 60xy 1 9y2 27. 1 }
4 a2 1
1 } 9 ab 1
1 } 81 b2
Solve the equation.
28. x2 2 9 5 0 29. p2 1 14p 1 49 5 0 30. d2 2 10d 1 25 5 0
ga1_ntg_02.indd 97ga1_ntg_02.indd 97 4/20/07 11:45:52 AM4/20/07 11:45:52 AM
98 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
31. 25m2 2 1 5 0 32. r2 2 2r 1 1 5 0 33. n2 1 20n 1 100 5 0
34. 4y2 2 9 5 0 35. 36x2 2 64 5 0 36. w2 1 4w 1 4 5 0
37. Washers Washers are available in many different sizes.
x
y a. Write and factor an expression for the area of one side
of the washer. Leave your answer in terms of π.
b. Find the area of one side of the washer when x 5 8 centimeters and y 5 3 centimeters.
38. Cherry Tree A cherry falls from a tree branch that is 9 feet above the ground.
a. How far above the ground is the cherry after 0.2 second?
b. After how many seconds does the cherry reach the ground?
39. Wind Chime A wind chime falls from a roof that is 25 feet above the ground.
a. How far above the ground is the wind chime after 0.5 second?
b. After how many seconds does the wind chime reach the ground?
LESSON
2.8 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 98ga1_ntg_02.indd 98 4/20/07 11:45:53 AM4/20/07 11:45:53 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 99
2.9 Factor Polynomials CompletelyGoal p Factor polynomials completely.Georgia
PerformanceStandard(s)
MM1A2f
Your Notes
VOCABULARY
Factor by grouping
Factor completely
Factor the expression.
a. 3x(x 1 2) 2 2(x 1 2) b. y2(y 2 4) 1 3(4 2 y)
Solution
a. 3x(x 1 2) 2 2(x 1 2) 5 (x 1 2)( )
b. The binomials y 2 4 and 4 2 y are .Factor from 4 2 y to obtain a common binomial factor.
y2(y 2 4) 1 3(4 2 y) 5 y2(y 2 4)
5 (y 2 4)
Example 1 Factor out common binomial
Factor the polynomial.
a. y3 1 7y2 1 2y 1 14 b. y2 1 2y 1 yx 1 2x
Solution
a. y3 1 7y2 1 2y 1 14 5 ( ) 1 ( ) 5 ( ) 1 ( ) 5 ( )( )
b. y2 1 2y 1 yx 1 2x 5 ( ) 1 ( )
5 ( ) 1 ( )
5 ( )( )
Example 2 Factor by grouping
Remember that you can check a factorization by multiplying the factors.
ga1nt_02.indd 56 4/16/07 9:24:53 AM
ga1_ntg_02.indd 99ga1_ntg_02.indd 99 4/20/07 11:45:53 AM4/20/07 11:45:53 AM
100 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Factor x3 2 12 1 3x 2 4x2.
Solution
x3 2 12 1 3x 2 4x2 5
5
5
5
Example 3 Factor by grouping
1. 5z(z 2 6) 1 4(z 2 6) 2. 2y2(y 2 1) 1 7(1 2 y)
3. x3 2 4x2 1 5x 2 20 4. n3 1 48 1 6n 1 8n2
Checkpoint Factor the expression.
Factor the polynomial completely.
a. x2 1 3x 2 1
b. 3r3 2 21r2 1 30r
c. 9d4 2 4d2
Solution
a. This polynomial be factored.
b. 3r3 2 21r2 1 30r 5
5
c. 9d4 2 4d2 5
5
Example 4 Factor completely
Rearrange the terms so that you can group terms within a common factor.
ga1nt_02.indd 57 4/16/07 9:24:54 AM
ga1_ntg_02.indd 100ga1_ntg_02.indd 100 4/20/07 11:45:55 AM4/20/07 11:45:55 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 101
Your Notes
5. 22x3 1 6x2 1 108x 6. 12y4 2 75y2
Checkpoint Factor the expression.
7. 2x3 1 2x2 5 40x 8. 24x3 1 72x 5 212x2
Checkpoint Solve the equation.
Homework
Solve 5x3 2 25x2 5 230x.
Solution
5x3 2 25x2 5 230x Write original equation.
5x3 2 25x2 30x 5 0 30xto each side.
5 0 Factor out .
5 0 Factor trinomial.
or or Zero-productproperty
x 5 x 5 x 5 Solve for x.
Example 5 Solve a polynomial equation
Remember that you can check your answers by substituting each solution for x in the original equation.
ga1nt_02.indd 58 4/16/07 9:24:56 AM
ga1_ntg_02.indd 101ga1_ntg_02.indd 101 4/20/07 11:45:56 AM4/20/07 11:45:56 AM
102 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Match the trinomial with its correct factorization.
1. 2x(x 1 5) 2 (x 1 5) 2. 2x(x 1 5) 1 (x 1 5) 3. 2x(x 2 5) 2 (x 2 5)
A. (2x 1 1)(x 1 5) B. (2x 2 1)(x 2 5) C. (2x 2 1)(x 1 5)
Factor the expression.
4. x(x 1 4) 1 (x 1 4) 5. b(b 1 3) 2 (b 1 3) 6. 2m(m 1 1) 1 (m 1 1)
7. 5r(r 1 2) 2 (r 1 2) 8. w(w 1 6) 1 3(w 1 6) 9. y(y 1 4) 2 6(y 1 4)
10. n(n 2 3) 2 7(n 2 3) 11. 3z(z 2 4) 1 8(z 2 4) 12. 2p(p 1 5) 2 3(p 1 5)
Factor the polynomial by grouping.
13. x2 1 x 1 3x 1 3 14. x2 2 x 1 2x 2 2 15. x2 1 8x 2 x 2 8
16. x3 2 5x2 1 2x 2 10 17. x3 2 4x2 2 6x 1 24 18. x3 1 3x2 1 5x 1 15
19. x3 2 x2 1 7x 2 7 20. x3 1 3x2 2 3x 2 9 21. x3 1 3x2 2 x 2 3
Determine whether the polynomial has been completely factored.
22. x4 1 x3 23. x2 1 1 24. 2x2 1 4
LESSON
2.9 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 102ga1_ntg_02.indd 102 4/20/07 11:45:58 AM4/20/07 11:45:58 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 103
Name ——————————————————————— Date ————————————
LESSON
2.9 Practice continued
Factor the polynomial completely.
25. x5 2 x3 26. 4a4 2 25a2 27. 5y6 2 125y4
Solve the equation.
28. x3 1 x2 2 25x 2 25 5 0 29. x3 1 x2 2 16x 2 16 5 0 30. x3 2 x2 2 4x 1 4 5 0
31. x3 2 x2 2 9x 1 9 5 0 32. z3 2 4z 5 0 33. c4 2 64c2 5 0
34. Metal Plate You have a metal plate that you have drilled a hole into. The entire area enclosed by the metal plate is given by 5x2 1 12x 1 10 and the area of the hole is given by x2 1 2. Write an expression for the area in factored form of the plate that is left after the hole is drilled.
35. Storage Container A plastic storage container in the shape of a cylinder has a height of 8 inches and a volume of 72π cubic inches.
a. Write an equation for the volume of the storage container.
b. What is the radius of the storage container?
36. Tennis Ball For a science experiment, you toss a tennis ball from a height of 32 feet with an initial upward velocity of 16 feet per second. How long will it take the tennis ball to reach the ground?
ga1_ntg_02.indd 103ga1_ntg_02.indd 103 4/20/07 11:45:59 AM4/20/07 11:45:59 AM
104 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.10 Graph y 5 ax2 1 c
Goal p Graph the simple quadratic functions.GeorgiaPerformanceStandard(s)
MM1A1b, MM1A1c, MM1A1e
Your Notes
VOCABULARY
Quadratic function
Parabola
Parent quadratic function
Vertex
Axis of Symmetry
PARENT QUADRATIC FUNCTION
The most basic quadratic function in the family of quadratic functions, called the
, is y 5 x2. The graph is shown below.
The line that passes through
x
y
1
3
5
12121
23 3
(0, 0)
x 5 0y 5 x2
the vertex and divides the parabola into two symmetric parts is called the
. The axis of symmetry for the graph of y 5 x2 is the y-axis, .
The lowest or highest point on the parabola is the . The vertex of the graph of y 5 x2
is ( , ).
ga1nt_02.indd 59 4/16/07 9:24:57 AM
ga1_ntg_02.indd 104ga1_ntg_02.indd 104 4/20/07 11:45:59 AM4/20/07 11:45:59 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 105
Your Notes
1. y 5 25x2
x
y3
121
29
23
215
221
23 3
Checkpoint Graph the function. Compare the graph with the graph of y 5 x2.
Graph y 51}2
x2. Compare the graph with the graph of
y 5 x2.
SolutionStep 1 Make a table of values
x
y
1
3
5
7
12123 3
for y 51}2 x2.
x 24 22 0 2 4
y
Step 2 the points from the table.
Step 3 Draw a through the points.
Step 4 Compare the graphs of y 51}2 x2 and y 5 x2. Both
graphs have the same vertex, ( , ), and axis of symmetry, . However, the graph of
y 51}2 x2 is than the graph of y 5 x2. This
is because the graph of y 51}2 x2 is a vertical
1by a factor of 2 of the graph of
y 5 x2.
Example 1 Graph y 5 ax2
ga1nt_02.indd 60 4/16/07 9:24:57 AM
ga1_ntg_02.indd 105ga1_ntg_02.indd 105 4/20/07 11:46:01 AM4/20/07 11:46:01 AM
106 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
The stained glass window shown
x
y
2
14
22226 6
6
10
can be modeled by the graph of the function y 5 20.25x2 1 16 where x and y are measured in inches. Find the domain and range of the function in this situation.
Solution
Step 1 Find the domain. In the graph, the window extends inches on either side of the origin. So the domain is .
Step 2 Find the range using the fact that the highest point on the window is ( , ) and the lowest point, , occurs at each end.
y 5 20.25( )2 1 16 5 , so the range is .
Example 3 Use a graph
Graph y 5 23x2 1 3. Compare the graph with the graph of y 5 x2.
Step 1 Make a table of values for
x
y2
12122
26
210
23 3y 5 23x2 1 3.
x 22 21 0 1 2
y
Step 2 the points from the table.
Step 3 Draw a through the points.
Step 4 Compare the graphs. Both graphs have the same axis of symmetry. However, the graph of y 5 23x2 1 3 is and has a vertex than the graph of y 5 x2 because the graph of y 5 23x2 1 3 is a and a of the graph of y 5 x2.
Example 2 Graph y 5 ax2 1 c
ga1nt_02.indd 61 4/16/07 9:24:59 AM
ga1_ntg_02.indd 106ga1_ntg_02.indd 106 4/20/07 11:46:02 AM4/20/07 11:46:02 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 107
Your Notes
2. y 51}4 x2 2 6
x
y
1
12121
23
25
23 3
3. y 5 2x2 2 3
x
y
1
12121
23
25
23 3
4. y 5 2 8x2 1 5
x
y
1
3
5
7
12121
23 3
Checkpoint Graph the function. Compare the graph with the graph of y 5 x2.
5. In Example 3, suppose the
x
y
2
14
22222
26 6
6
10
stained glass window is modeled by the function y 5 20.25x2 1 9. Find the domain and range in this situation.
Checkpoint Complete the following exercise.
Homework
ga1nt_02.indd 62 4/16/07 9:25:00 AM
ga1_ntg_02.indd 107ga1_ntg_02.indd 107 4/20/07 11:46:04 AM4/20/07 11:46:04 AM
108 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Use the quadratic function to complete the table of values.
1. y 5 5x2 2. y 5 24x2
x 22 21 0 1 2
y
x 22 21 0 1 2
y
3. y 5 x2 1 6 4. y 5 x2 2 8
x 22 21 0 1 2
y
x 22 21 0 1 2
y
Match the function with its graph.
5. y 5 2 1 } 2 x2 6. y 5 2x2 7. y 5
1 } 4 x2
A.
x
y
1
3
12121
23 3
B.
x
y
1
3
12121
23 3
C.
x
y
1
21
23
23 3
Graph the function and identify its domain and range. Compare the graph with the graph of y 5 x2.
8. y 5 5x2 9. y 5 2 1 } 3 x2 10. y 5 26x2
x
y
1
3
5
12121
23 3
x
y
1
12121
23
23 3
x
y
12121
23
25
23 3
LESSON
2.10 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 108ga1_ntg_02.indd 108 4/20/07 11:46:06 AM4/20/07 11:46:06 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 109
Name ——————————————————————— Date ————————————
LESSON
2.10 Practice continued
Identify the vertex and axis of symmetry of the graph.
11.
x
y
2
6
10
12123 3
12. x
y
12121
23
23 3
13.
x
y
0.5
123 3
Match the function with its graph.
14. y 5 x2 2 3 15. y 5 3x2 2 1 16. y 5 2x2 1 3
A.
x
y
123 3
1
21
B.
x
y
1
12121
23 3
C.
x
y
1
3
12123 3
Graph the function and identify its domain and range. Compare the graph with the graph of y 5 x2.
17. y 5 x2 2 5 18. y 5 x2 1 7 19. y 5 2x2 2 3
x
y1
12121
23
25
23 3
x
y
2
6
10
12122
23 3
x
y
1
3
12121
23
23 3
ga1_ntg_02.indd 109ga1_ntg_02.indd 109 4/20/07 11:46:08 AM4/20/07 11:46:08 AM
110 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Complete the statement.
20. The graph of y 5 x2 1 5 can be obtained from the graph of y 5 x2 by shifting the graph of y 5 x2 ? .
21. The graph of y 5 10x2 can be obtained from the graph of y 5 x2 by ? the graph of y 5 x2 by a factor of ? .
22. Pot Rack A cross section of the pot rack shown can be modeled
x
y
2
6
22226 6
by the graph of the function y 5 20.08x2 1 8 where x and y are measured in inches.
a. Find the domain of the function in this situation.
b. Find the range of the function in this situation.
23. Drawer Handle A cross section of the drawer handle shown
x
y
1
3
1212325 3 5
can be modeled by the graph of the function y 5 2 1 } 18 x2 1 2
where x and y are measured in centimeters.
a. Find the domain of the function in this situation.
b. Find the range of the function in this situation.
LESSON
2.10 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 110ga1_ntg_02.indd 110 4/20/07 11:46:10 AM4/20/07 11:46:10 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 111
2.11 Graph y 5 ax2 1 bx 1 c
Goal p Graph general quadratic functions.GeorgiaPerformanceStandard(s)
MM1A1a, MM1A1c, MM1A1d
Your Notes
VOCABULARY
Minimum value
Maximum value
PROPERTIES OF THE GRAPH OF A QUADRATIC FUNCTION
The graph of y 5 ax2 1 bx 1 c is a parabola that:
• opens if a > 0 and opens if a < 0.
• is narrower than the graph of y 5 x2 if⏐a⏐ 1 and wider if⏐a⏐ 1.
• has an axis of symmetry of
x
y
(0, c)
x 5 2b2a
x 5 .
• has a vertex with an
x-coordinate of .
• has a y-intercept of .So, the point ( , ) is on the parabola.
MINIMUM AND MAXIMUM VALUES
For y 5 ax2 1 bx 1 c, the y-coordinate of the vertex is the value of the function if a 0 and the value of the function if a 0.
ga1nt_02.indd 63 4/16/07 9:25:01 AM
ga1_ntg_02.indd 111ga1_ntg_02.indd 111 4/20/07 11:46:11 AM4/20/07 11:46:11 AM
112 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Tell whether the function f(x) 5 5x2 2 20x 1 17 has a minimum value or a maximum value. Then find the minimum or maximum value.
Solution
Because a 5 and , the parabola opens and the function has a value. To find the value, find the .
x 5 2b
}2a 5 5 The x-coordinate is
2b
}2a.
f( ) 5 5( )2 2 20( ) 1 17 Substitute for x.
5 Simplify.
The value of the function is .
Example 2 Find the minimum or maximum value
Consider the function y 5 22x2 1 16x 2 15.
a. Find the axis of symmetry of the graph of the function.
b. Find the vertex of the graph of the function.
Solution
a. For the function y 5 22x2 1 16x 2 15, a 5and b 5 .
x 5 2b
}2a 5 5
The axis of symmetry is x 5 .
b. The x-coordinate of the vertex is 2b
}2a, or . To find
the y-coordinate, substitute for x in the function and find y.
y 5 22( )2 1 16( ) 2 15 5
The vertex is ( , ).
Example 1 Find the axis of symmetry and the vertex
ga1nt_02.indd 64 4/16/07 9:25:02 AM
ga1_ntg_02.indd 112ga1_ntg_02.indd 112 4/20/07 11:46:12 AM4/20/07 11:46:12 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 113
Your Notes
3. Tell whether the function f(x) 5 21}2 x2 1 6x 1 8 has
a minimum value or a maximum value. Then find the minimum or maximum value.
Checkpoint Complete the following exercise.
1. y 5 3x2 1 18x 1 5
2. y 51}4 x2 2 4x 1 7
Checkpoint Find the axis of symmetry and the vertex of the graph of the function.
ga1nt_02.indd 65 4/16/07 9:25:03 AM
ga1_ntg_02.indd 113ga1_ntg_02.indd 113 4/20/07 11:46:14 AM4/20/07 11:46:14 AM
114 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
4. Graph the function
x
y
3
9
15
21
12123
23 3
y 5 4x2 1 8x 1 3. Label the vertex and axis of symmetry.
Checkpoint Complete the following exercise.
Graph y 5 2x2 1 4x 2 1.
Step 1 Determine whether the parabola opens up or down. Because a 0, the parabola opens .
Step 2 Find and draw the axis of
x
y
1
3
12121
23
3 5
symmetry:
x 5 2b
}2a 5 5 .
Step 3 Find and plot the vertex. The x-coordinate of the vertex is , or .
To find the y-coordinate, substitute for x in the function and simplify.
y 5 2( )2 1 4( ) 2 1 5 3
So, the vertex is ( , ).
Step 4 Plot two points. Choose x 1 0
ytwo x-values less than the x-coordinate of the vertex. Then find the corresponding y-values.
Step 5 the points plotted in Step 4 in the axis of symmetry.
Step 6 Draw a through the plotted points.
Example 3 Graph y 5 ax2 1 bx 1 c
ga1nt_02.indd 66 4/16/07 9:25:04 AM
ga1_ntg_02.indd 114ga1_ntg_02.indd 114 4/20/07 11:46:15 AM4/20/07 11:46:15 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 115
Your Notes
Compare the graph of f(x) 5 x2 2 8x 1 16 and g(x) 5 2x2 1 8x 2 16.
Solution
x
y
1
3
12121
23
7
f (x) 5 x2 2 8x 1 16
g(x) 5 2x2 1 8x 2 16
Consider the as a mirror. The graph of (x) 5 2x2 1 8x 2 16 is the mirror image of the
graph of f(x) 5 x2 2 8x 1 16. So, the graph of g(x)is a of the graph of f(x).
Example 4 Compare graphs
5. f(x) 5 x2 1 6x 1 5
x
y
2 62226
23
g(x) 5 x2 2 6x 1 5f (x) 5 x2 1 6x 1 5
g(x) 5 x2 2 6x 1 5
6. f(x) 5 x2 1 3x 1 4
x
y
1
12123252721
f (x) 5 x2 1 3x 1 4
g(x) 5 2x2 2 3x 2 4
g(x) 5 2x2 2 3x 2 4
Checkpoint Compare the graphs of f(x) and g(x).
Homework
ga1nt_02.indd 67 4/16/07 9:25:05 AM
ga1_ntg_02.indd 115ga1_ntg_02.indd 115 4/20/07 11:46:16 AM4/20/07 11:46:16 AM
116 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Identify the values of a, b, and c in the quadratic function.
1. y 5 7x2 1 2x 1 11 2. y 5 3x2 2 5x 1 1 3. y 5 4x2 1 2x 2 2
4. y 5 23x2 1 9x 1 4 5. y 5 1 } 2 x2 2 x 2 5 6. y 5 2x2 1 7x 2 6
Tell whether the graph opens upward or downward. Then fi nd the axis of symmetry of the graph of the function.
7. y 5 x2 1 6 8. y 5 2x2 2 1 9. y 5 x2 1 6x 1 1
10. y 5 x2 2 4x 1 5 11. y 5 2x2 1 4x 2 5 12. y 5 2x2 1 8x 1 3
13. y 5 x2 1 3x 2 6 14. y 5 2x2 1 7x 2 2 15. y 5 3x2 1 6x 1 10
Find the vertex of the graph of the function.
16. y 5 x2 1 5 17. y 5 2x2 1 3 18. y 5 x2 1 10x 1 3
LESSON
2.11 Practice
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 116ga1_ntg_02.indd 116 4/20/07 11:46:18 AM4/20/07 11:46:18 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 117
Name ——————————————————————— Date ————————————
LESSON
2.11 Practice continued
19. y 5 2x2 1 4x 2 2 20. y 5 3x2 1 6x 1 1 21. y 5 22x2 1 8x 2 3
22. y 5 10x2 2 10x 1 7 23. y 5 x2 1 x 1 3 24. y 5 x2 2 x 1 1
Use the quadratic function to complete the table of values.
25. y 5 x2 2 6x 1 8 26. y 5 2x2 1 12x 2 5
x 1 2 3 4 5
y
x 4 5 6 7 8
y
27. y 5 7x2 1 14x 1 2 28. y 5 22x2 2 4x 1 1
x 23 22 21 0 1
y
x 23 22 21 0 1
y
Match the function with its graph.
29. y 5 8x2 1 2x 1 3 30. y 5 2x2 1 8x 1 1 31. y 5 1 } 2 x2 1 8x 1 5
A.
x
y4
424220
B. x
y
121
27
2325
C.
x
y
1
12123 3
ga1_ntg_02.indd 117ga1_ntg_02.indd 117 4/20/07 11:46:18 AM4/20/07 11:46:18 AM
118 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Graph the function. Label the vertex and axis of symmetry.
32. y 5 2x2 2 6 33. y 5 x2 1 7 34. y 5 x2 1 2x 1 5
x
y2
12122
26
210
23 3
x
y
2
6
10
12122
23 3
x
y
1
3
5
7
1212325 3
35. y 5 x2 2 8x 1 1 36. y 5 22x2 1 x 2 3 37. y 5 2x2 2 4x 1 3
x
y
2 6 102222
26
210
214
x
y
12121
23
25
27
23 3
x
y
1
3
5
7
1212325
Tell whether the function has a minimum value or a maximum value. Then fi nd the minimum or maximum value.
38. f (x) 5 x2 2 7 39. f (x) 5 2x2 1 9 40. f (x) 5 2x2 1 4x
41. f (x) 5 2x2 1 2x 2 3 42. f (x) 5 1 } 4 x2 2 8x 1 1 43. f (x) 5 23x2 1 11
LESSON
2.11 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 118ga1_ntg_02.indd 118 4/20/07 11:46:20 AM4/20/07 11:46:20 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 119
Name ——————————————————————— Date ————————————
LESSON
2.11 Practice continued
Compare the graphs of f (x) and g(x).
44.
x
y
1
22226 6
f (x) 5 x2 1 8x 1 16
g(x) 5 x2 2 8x 1 16
45.
x
y
21
23
23 21 1
f (x) 5 2x2 2 4x 1 3
g(x) 5 22x2 1 4x 2 3
46.
x
y
1
2325
g(x) 5 2x2 2 4x 2 4
f (x) 5 x2 1 4x 1 4
47. Greenhouse The dome of the greenhouse shown can be modeled by the graph of the function y 5 20.15625x2 1 2.5x where x and y are measured in feet. What is the height h at the highest point of the dome as shown in the diagram?
x
y
2
6
10
2 6 10 14
h
48. Fencing A parabola forms the top of a fencing panel as shown. This parabola can be modeled by the graph of the function y 5 0.03125x2 2 0.25x 1 4 where x and y are measured in feet and y represents the number of feet the parabola is above the ground. How far above the ground is the lowest point of the parabola formed by the fence?
ga1_ntg_02.indd 119ga1_ntg_02.indd 119 4/20/07 11:46:21 AM4/20/07 11:46:21 AM
120 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.12 Solve Quadratic Equations by GraphingGoal p Solve quadratic equations by graphing.Georgia
PerformanceStandard(s)
MM1A1d, MM1A3c
Your Notes
VOCABULARY
Quadratic equation
Solve 2x2 1 2x 5 28 by graphing.
Step 1 Write the equation in .
2x2 1 2x 5 28 Write original equation.
2x2 1 2x 1 8 5 Add to each side.
Step 2 Graph the function y 5 2x2 1 2x 1 8. The x-intercepts are and .
x
y
2
6
10
14
12123 3
The solutions of the equation 2x2 1 2x 5 28 are and .
CHECK You can check and in the original equation.
2x2 1 2x 5 28 2x2 1 2x 5 28
2( )2 1 2( ) 0 28 2( )2 1 2( ) 0 28
5 5
Example 1 Solve a quadratic equation having two solutions
ga1nt_02.indd 68 4/16/07 9:25:06 AM
ga1_ntg_02.indd 120ga1_ntg_02.indd 120 4/20/07 11:46:23 AM4/20/07 11:46:23 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 121
Your Notes
Solve x2 2 4x 5 24 by graphing.
x
y
1
3
5
7
12123 3
Step 1 Write the equation in standard form.
x2 2 4x 5 24 Write original equation.
x2 2 4x 1 4 5 Add to each side.
Step 2 the function y 5 x2 2 4x 1 4. The x-intercept is .
The solution of the equation x2 2 4x 5 24 is .
Example 2 Solve a quadratic equation having one solution
Solve x2 1 8 5 2x by graphing.
x
y
2
6
10
14
12123 3
Step 1 Write the equation in standard form.
x2 1 8 5 2x Write original equation.
Subtract from each side.
Step 2 the function y 5 .The graph has x-intercepts.
The equation x2 1 8 5 2x has .
Example 3 Solve a quadratic equation having no solution
1. Solve x2 2 6 5 25x
x
y
3
9
12123
29
2325
by graphing.
Checkpoint Complete the following exercise.
ga1nt_02.indd 69 4/16/07 9:25:07 AM
ga1_ntg_02.indd 121ga1_ntg_02.indd 121 4/20/07 11:46:25 AM4/20/07 11:46:25 AM
122 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your Notes
Find the zeros of f(x) 5 2x2 2 8x 2 7.
Graph the function
x
y
3
9
22223
29
26 6
f(x) 5 2x2 2 8x 2 7. Thex-intercepts are and .
The zeros of the function are and .
CHECK Substitute and in the original function.
f( ) 5 2( )2 2 8( ) 2 7 5
f( ) 5 2( )2 2 8( ) 2 7 5
Example 4 Find the zeros of a quadratic function
2. x2 1 9 5 6x
x
y
1
3
5
121 3 521
3. x2 2 7x 5 215
x
y
1
3
5
1 3 5
Checkpoint Solve the quadratic equation by graphing.
4. f(x) 5 2x2 1 6x 2 5
x
y
1
3
1 3 5
Checkpoint Find the zeros of the function.
Homework
ga1nt_02.indd 70 4/18/07 1:57:03 PM
ga1_ntg_02.indd 122ga1_ntg_02.indd 122 4/20/07 11:46:27 AM4/20/07 11:46:27 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 123
Name ——————————————————————— Date ————————————
LESSON
2.12 PracticeWrite the equation in standard form.
1. x2 1 3x 5 212 2. x2 2 8x 5 14 3. x2 5 9x 2 1
4. x2 5 6 2 10x 5. 14 2 x2 5 3x 6. 1 }
2 x2 5 23x 2 7
Determine whether the given value is a solution of the equation.
7. x2 1 36 5 0; 26 8. 100 2 x2 5 0; 210 9. 0 5 x2 1 6x 1 5; 21
10. x2 2 5x 1 6 5 0; 2 11. 2x2 1 4x 2 4 5 0; 4 12. 0 5 2x2 1 8x 1 3; 8
Use the graph to fi nd the solutions of the given equation.
13. x2 1 5 5 0 14. 2x2 1 4 5 0 15. x2 1 4x 1 3 5 0
x
y
1
3
12123 3
x
y
1
3
5
12123 3
x
y
3
121
25
ga1_ntg_02.indd 123ga1_ntg_02.indd 123 4/20/07 11:46:29 AM4/20/07 11:46:29 AM
124 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
16. x2 2 16 5 0 17. x2 2 2 5 0 18. x2 1 2x 2 8 5 0
x
y
4
212 12
x
y1
12123 321
25
x
y
2
2222
26
26
Solve the equation by graphing.
19. 8x2 1 2x 1 3 5 0 20. 2x2 1 3x 1 1 5 0 21. 1 } 2 x2 1 4x 1 6 5 0
x
y
3
9
15
12123 3
x
y
1
3
5
12123 3
x
y
2
6
22222
26
22. x2 2 2x 2 15 5 0 23. 22x2 1 x 2 3 5 0 24. 2x2 2 2x 1 3 5 0
x
y3
32323
29
215
29 9
x
y3
12123
29
215
23 3
x
y
1
3
12121
23
LESSON
2.12 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 124ga1_ntg_02.indd 124 4/20/07 11:46:30 AM4/20/07 11:46:30 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 125
Name ——————————————————————— Date ————————————
LESSON
2.12 Practice continued
Find the zeros of the function by graphing the function.
25. f (x) 5 x2 2 25 26. f (x) 5 2x2 1 9 27. f (x) 5 2x2 1 4x
x
y5
52525
215
225
215 15
x
y
3
9
32323
29
29 9
x
y
1
3
12121
23
23 3
28. f (x) 5 x2 2 4x 2 12 29. f (x) 5 2x2 2 3x 1 40 30. f (x) 5 3x2 2 30x
x
y4
22224
212
220
6
x
y
10
30
50
222210
26
x
y
222212
236
260
6 10
31. Plate Cover A plate cover made of netting has a cross section
00
2
4
6
8
10
12
4 8 12 162 6 10 14Width (inches)
He
igh
t (i
nch
es)
y
x
in the shape of a parabola. The cross section can be modeled by the function y 5 20.1875x2 1 3x where x is the width of the cover (in inches) and y is the height of the cover (in inches).
a. Graph the function.
b. Find the domain and range of the function in this situation.
c. How wide is the cover?
d. How tall is the cover?
ga1_ntg_02.indd 125ga1_ntg_02.indd 125 4/20/07 11:46:32 AM4/20/07 11:46:32 AM
126 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
2.13 Use Square Roots to Solve Quadratic EquationsGoal p Solve a quadratic equation by finding square roots.Georgia
PerformanceStandard(s)
MM1A3a
Your Notes
Solve the equation.
a. z2 2 5 5 4 b. r2 1 7 5 4
Solutiona. z2 2 5 5 4 Write original equation.
z2 5 Add to each side.
z 5 Take square roots of each side.
z 5 Simplify.
The solutions are and .
b. r2 1 7 5 4 Write original equation.
r2 5 Subtract from each side.
Negative real numbers do not have real .So, there is .
Example 1 Solve quadratic equations
VOCABULARY
Square root
Radicand
Perfect square
ga1nt_02.indd 71 4/16/07 9:25:10 AM
ga1_ntg_02.indd 126ga1_ntg_02.indd 126 4/20/07 11:46:33 AM4/20/07 11:46:33 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 127
Your Notes
1. 3x2 5 108 2. t2 1 17 5 17 3. 81p2 5 4
Checkpoint Solve the equation.
Solve 4x2 1 3 5 23. Round the solutions to the nearest hundredth.
Solution4x2 1 3 5 23 Write original equation.
4x2 5 Subtract from each side.
x2 5 Divide each side by .
x 5 Take square roots of each side.
x ø Use a calculator. Round to the nearest hundredth.
The solutions are about and .
Example 3 Approximate solutions of a quadratic equation
Solve the equation 25k2 5 9.
Solution25k2 5 9 Write original equation.
k2 5 Divide each side by .
k 5 Take square roots of each side.
k 5 Simplify.
The solutions are and .
Example 2 Take square root of a fraction
ga1nt_02.indd 72 4/16/07 9:25:11 AM
ga1_ntg_02.indd 127ga1_ntg_02.indd 127 4/20/07 11:46:34 AM4/20/07 11:46:34 AM
128 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Your NotesCheckpoint Solve the equation. Round the solutions to the nearest hundredth.
4. 2x2 2 7 5 9 5. 6g2 1 1 5 19
Solve 5(x 1 1)2 5 30. Round the solutions to the nearest hundredth.
Solution5(x 1 1)2 5 30 Write original equation.
(x 1 1)2 5 Divide each side by .
x 1 1 5 Take square roots of each side.
x 5 Subtract from each side.
The solutions are ø and
ø .
Example 4 Solve a quadratic equation
Homework
Checkpoint Solve the equation. Round the solutions to the nearest hundredth, if necessary.
6. 3(m 2 4)2 5 12 7. 4(a 2 3)2 5 32
ga1nt_02.indd 73 4/16/07 9:25:12 AM
ga1_ntg_02.indd 128ga1_ntg_02.indd 128 4/20/07 11:46:36 AM4/20/07 11:46:36 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 129
Name ——————————————————————— Date ————————————
LESSON
2.13 PracticeEvaluate the expression.
1. Ï}
49 2. Ï}
225 3. Ï}
100
Isolate the variable in the equation.
4. 9x2 2 18 5 0 5. 4x2 2 12 5 0 6. 10x2 2 40 5 0
Solve the equation.
7. x2 5 36 8. x2 2 9 5 0 9. 5x2 5 20
10. 5x2 2 45 5 0 11. 2x2 2 18 5 0 12. 3x2 2 12x 5 0
Evaluate the expression. Round your answer to the nearest hundredth.
13. Ï}
5 14. Ï}
10 15. Ï}
12
Solve the equation. Round the solutions to the nearest hundredth.
16. x2 5 8 17. x2 2 3 5 0 18. 7x2 2 14 5 0
ga1_ntg_02.indd 129ga1_ntg_02.indd 129 4/20/07 11:46:37 AM4/20/07 11:46:37 AM
130 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Use the given area A of the circle to fi nd the radius r or the diameter d of the circle. Round the answer to the nearest hundredth, if necessary.
19. A = 25π m2 20. A = 121π in.2 21. A = 23π cm2
r
r
d
22. Boat Racing The maximum speed s (in knots or nautical miles per hour) that some
kinds of boats can travel can be modeled by s2 5 16
} 9 x where x is the length of the
water line in feet. Find the maximum speed of a sailboat with a 20-foot water line. Round your answer to the nearest hundredth.
23. Tanks You can fi nd the radius r (in inches) of a cylindrical air compressor
receiver tank by using the formula c 5 1 } 73.53 hr2 where h is the height of the tank
(in inches) and c is the capacity of the tank (in gallons). Find the tank radius of each tank in the table. Round your answers to the nearest inch.
Tank Height (in.) Radius (in.) Capacity (in.3)
A 24 12
B 36 24
C 48 65
LESSON
2.13 Practice continued
Name ——————————————————————— Date ————————————
ga1_ntg_02.indd 130ga1_ntg_02.indd 130 4/20/07 11:46:38 AM4/20/07 11:46:38 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 131
Words to ReviewGive an example of the vocabulary word.
Monomial
Polynomial
Leading coefficient
Trinomial
Volume model for polynomial arithmetic
Degree of a monomial
Degree of a polynomial
Binomial
Area model for polynomial arithmetic
Pascal's Triangle
ga1nt_02.indd 74 4/16/07 9:25:13 AM
ga1_ntg_02.indd 131ga1_ntg_02.indd 131 4/20/07 11:46:39 AM4/20/07 11:46:39 AM
132 Georgia Notetaking Guide, Mathematics 1 Copyright © McDougal Littell/Houghton Mifflin Company.
Roots
Perfect square trinomial
Factor completely
Parabola
Vertical motion model
Factor by grouping
Quadratic function
Parent quadratic function
ga1nt_02.indd 75 4/16/07 9:25:14 AM
ga1_ntg_02.indd 132ga1_ntg_02.indd 132 4/20/07 11:46:40 AM4/20/07 11:46:40 AM
Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 1 133
Vertex
Minimum value
Quadratic equation
Radicand
Axis of symmetry
Maximum value
Square root
Perfect square
ga1nt_02.indd 76 4/16/07 9:25:14 AM
ga1_ntg_02.indd 133ga1_ntg_02.indd 133 4/20/07 11:46:41 AM4/20/07 11:46:41 AM